Machine learning on manifolds for inverse scattering: Lipschitz stability analysis

TL;DR

This paper develops Lipschitz stability estimates for neural network-based inverse maps on manifolds, enabling robust and accurate parameter recovery in physics-informed inverse scattering problems involving cracks.

Contribution

It introduces a novel theoretical framework for Lipschitz stability on manifolds in inverse scattering, with practical neural network applications for crack detection.

Findings

Neural network approximations achieve accurate inverse scattering reconstructions.

The method demonstrates robustness against variations in incident wave types and external forces.

Numerical experiments confirm the theoretical stability and efficiency of the approach.

Abstract

Establishing Lipschitz stability estimates is crucial for ensuring the mathematical robustness of neural network (NN) approximations in machine learning (ML)-based parameter estimation, particularly in physics-informed settings. In this work, we derive such estimates for the inverse of a nonlinear map defined on a manifold that captures both unknown parameters and the nonlinear physical processes they influence. Our analysis is based on finite-dimensional, learnable representations of the manifold and provides Lipschitz stability estimates on the manifold-based subspaces, for a class of inverse maps associated with parameter dependent linear compact operators. Such operators model scattered and far-field data that can be used to detect structures such as cracks. We apply our theoretical ML manifold framework to inverse Helmholtz problems in unbounded regions exterior to cracks, addressing the scattered-field data-driven inverse problem while ensuring injectivity conditions on the manifold, a requirement for the Lipschitz stability. Our method accurately recovers crack-defining parameters without requiring prior knowledge of inputs such as incident wave types or external forces on the crack. Numerical experiments using NN approximations confirm the accuracy, efficiency, and robustness of the proposed approach.